Refresher: Solving time-independent Schrödinger equation

\[ \begin{equation} \hat{H} \Psi(x) = \left[\frac{\hat{p}^2}{2m} + V(x)\right] \Psi(x) = E\Psi(x), \end{equation} \]

i.e. finding \(\Psi(x)\) as the eigenfunctions of Hamiltonian \(\hat{H}\).

In the position representation, the momentum operator \(\hat{p} = -i\hbar \mathrm{d}/\mathrm{d}x\), so we end up with an ODE:

\[ \begin{equation} \left[-\frac{\hbar^2}{2m}\frac{\mathrm{d}^2}{\mathrm{d}x^2} + V(x)- E \right] \Psi(x) = 0. \end{equation} \]

In piecewise-continuous potentials

\[\begin{equation} V(x) = \begin{cases} V_1,& \text{if } x< 0\\ V_2,& \text{if } x \geq 0, \end{cases} \end{equation} \]

In each domain \(i\)

\[ \begin{equation} \Bigg[\frac{\mathrm{d}^2}{\mathrm{d}x^2} + \underbrace{\frac{2m}{\hbar^2} (E - V_i)}_{k_i^2} \Bigg] \Psi_i(x) = 0 \end{equation} \]

If \(E>V_i\), we get \(k_i \in \mathbb{R}\).

General solutions with unknown amplitudes \(A_1,B_1,A_2,B_2\):

\[ \begin{align}\label{eq:solution1} \Psi_1(x) = \underbrace{A_1 e^{i k_1 x}}_{\rightarrow} + \underbrace{B_1 e^{-i k_1 x}}_{\leftarrow}, \quad \Psi_2(x) = \underbrace{A_2 e^{i k_2 x}}_{\rightarrow} + \underbrace{B_2 e^{-i k_2 x}}_{\leftarrow}, \end{align} \]

Boundary conditions:

\[ \begin{equation} \lim_{x\rightarrow 0} \Psi_1(x) = \lim_{x\rightarrow 0} \Psi_2(x), \end{equation} \]

\[ \begin{equation} \lim_{x\rightarrow 0} \frac{\mathrm{d}}{\mathrm{d}x}\Psi_1(x) = \lim_{x\rightarrow 0} \frac{\mathrm{d}}{\mathrm{d}x}\Psi_2(x), \end{equation} \]

(4 variables, 2 equations) - ill-defined problem?

• problem is linear, i.e. can set \(A_1=1\),
• particle incident from the left — no left-propagating component in \(i=2\):

\[ \begin{equation} \Psi_1(x) = \color{blue}{e^{i k_1 x}} + \color{orange}{B_1 e^{-i k_1 x}},\quad \Psi_2(x) = \color{green}{A_2 e^{i k_2 x}}, \end{equation} \]

add time evolution: \(\Psi(x,t) = e^{-iEt/\hbar}\Psi(x,0)\)

code available at github

problems

• what’s causing the oscillating behaviour of the wavefunction in the left part?
• separate the incident and reflected waves, plot or animate

what if \(V(x)>E\) somewhere?

• classical “hard border” (no penetration)

• quantum transmission

\[ \begin{equation} \Bigg[\frac{\mathrm{d}^2}{\mathrm{d}x^2} + \underbrace{\frac{2m}{\hbar^2} E}_{k_1^2} \Bigg] \Psi_1(x) = 0 \rightarrow \Psi_1(x) = \underbrace{A_1 e^{i k_1 x}}_{\rightarrow} + \underbrace{B_1 e^{-i k_1 x}}_{\leftarrow}, \end{equation} \]

\[ \begin{equation} \Bigg[\frac{\mathrm{d}^2}{\mathrm{d}x^2} + \underbrace{\frac{2m}{\hbar^2} E}_{k_3^2} \Bigg] \Psi_3(x) = 0 \rightarrow \Psi_3(x) = \underbrace{A_3 e^{i k_3 x}}_{\rightarrow} + \underbrace{B_3 e^{-i k_3 x}}_{\leftarrow}, \end{equation} \]

\[ \begin{equation} \Bigg[\frac{\mathrm{d}^2}{\mathrm{d}x^2} + \underbrace{\frac{2m}{\hbar^2} (E - V_2)}_{-\kappa_2^2} \Bigg] \Psi_2(x) = 0 \rightarrow \Psi_2(x) = \underbrace{B_2 e^{-\kappa_2 x}}_{\searrow} + \underbrace{A_2 e^{\kappa_2 x}}_{\nearrow}. \end{equation} \]

\[ \begin{equation} \Psi_2(x) = \color{orange}{\underbrace{B_2 e^{-\kappa_2 x}}_{\searrow}} + \color{blue}{\underbrace{A_2 e^{\kappa_2 x}}_{\nearrow}}. \end{equation} \]

Stitching solutions

\[ \begin{align} &\Psi_1(0) = \Psi_2(0),\quad \Psi_1'(0) = \Psi_2'(0)\\ &\Psi_2(L) = \Psi_3(L),\quad \Psi_2'(L) = \Psi_3'(L) \end{align} \]

6 variables \((A_1, B_1, A_2, B_2, A_3, B_3)\) but only 4 equations! note: no left-propagating fields in domain 3 (\(B_3=0\)), linear (\(A_1=1\)):

\[ \begin{align} &\Psi_1(x) = \color{blue}{\underbrace{e^{i k_1 x}}_{\rightarrow}} + \color{orange}{\underbrace{B_1 e^{-i k_1 x}}_{\leftarrow}},\\ &\Psi_2(x) = \underbrace{B_2 e^{-\kappa_2 x}}_{\searrow} + \underbrace{A_2 e^{\kappa_2 x}}_{\nearrow},\\ &\Psi_3(x) = \color{green}{\underbrace{A_3 e^{i k_3 x}}_{\rightarrow}}, \end{align} \]

Time-independent Schrodinger equation \(\rightarrow\) ODE: \[ \begin{equation} \left[-\frac{\hbar^2}{2m}\frac{\mathrm{d}^2}{\mathrm{d}x^2} + V(x)\right] \Psi(x) = E \Psi(x) \rightarrow \left[\frac{\mathrm{d}^2}{\mathrm{d}x^2} + \frac{2m}{\hbar^2} (E - V_i) \right] \Psi_i(x) = 0. \end{equation} \]

• potential step — \(V(x)<E\) everywhere

• potential barrier — \(V_b>E\)

potential barrier — \(V_b>E\)

boundary conditions:

• continuity of wavefunction at \(x=0\): \(\Psi_1(0) = \Psi_2(0)\)

• continuity of wavefunction derivative at \(x=0\): \(\Psi_1'(0) = \Psi_2'(0)\),

• continuity of wavefunction at \(x=L\): \(\Psi_2(L) = \Psi_3(L)\)

• continuity of wavefunction at derivative at \(x=L\): \(\Psi_2'(L) = \Psi_3'(L)\),

from homework: \[ \begin{align} &B_1 = \frac{(k_1-k_3)\kappa_2 \cosh(L \kappa_2) - i(k_1 k_3+\kappa_2^2)\sinh(L \kappa_2)}{(k_1+k_3)\kappa_2 \cosh(L \kappa_2) + i(- k_1 k_3+\kappa_2^2)\sinh(L \kappa_2)},\\ &A_3 = \frac{2 e^{-ik_3 L}k_1 \kappa_2}{(k_1+k_3)\kappa_2 \cosh(L \kappa_2) + i(- k_1 k_3+\kappa_2^2)\sinh(L \kappa_2)}, \end{align} \]

with \(k_1 = k_3 = \sqrt{2m E/\hbar^2},\) and \(\kappa_2 = \sqrt{2m(V_b - E)/\hbar^2}\)

derive with Wolfram Mathematica

M = {{
        1,-1,-1,0},
        {-\[ImaginaryI],-\[Kappa]2/k1,\[Kappa]2/k1,0},
        {0,Exp[\[Kappa]2 L],Exp[-\[Kappa]2 L],-Exp[\[ImaginaryI] k3 L]},
   {0,Exp[\[Kappa]2 L],- Exp[-\[Kappa]2 L],-\[ImaginaryI] k3/\[Kappa]2 Exp[\[ImaginaryI] k3 L]}
};
b = {-1, -\[ImaginaryI],0,0};
ln=LinearSolve[M,b];
B1 = ln[[1]]//FullSimplify
A2 = ln[[2]]//FullSimplify
B2 = ln[[3]]//FullSimplify
A3 = ln[[4]]//FullSimplify

wavefunctions \(\Psi(x)\) incident on a potential barrier

• \(V_b = 4\),
• \(E = 2\),
• \(L \kappa_2 = 0.3\).

wavefunctions \(\Psi(x, t) = \exp(-iE t/\hbar)\Psi(x)\) incident on a potential barrier

• \(V_b = 4\),
• \(E = 2\),
• \(L \kappa_2 = 0.3\).

wavefunctions \(\Psi(x, t) = \exp(-iE t/\hbar)\Psi(x)\) incident on a potential barrier

• \(V_b = 4\),
• \(E = 2\),
• \(L \kappa_2 = 1\).

wavefunctions \(\Psi(x, t) = \exp(-iE t/\hbar)\Psi(x)\) incident on a potential barrier

• \(V_b = 4\),
• \(E = 2\),
• \(L \kappa_2 = 2\).

wavefunctions \(\Psi(x, t) = \exp(-iE t/\hbar)\Psi(x)\) incident on a potential barrier

• \(V_b = 4\),
• \(E = 2\),
• \(L \kappa_2 = 8\).

transmission \(\mathcal{T}\) and reflection \(\mathcal{R}\) can be calculated from the current densities \(J_i = \frac{\hbar}{m} \Im (\Psi^* \Psi')\) (see ref…):

\[ \begin{align} &\color{orange}{\mathcal{R} = \frac{J_R}{J_I} = |B_1|^2} = \left|\frac{(k_1-k_3)\kappa_2 \cosh(L \kappa_2) - i(k_1 k_3+\kappa_2^2)\sinh(L \kappa_2)}{(k_1+k_3)\kappa_2 \cosh(L \kappa_2) + i(- k_1 k_3+\kappa_2^2)\sinh(L \kappa_2)}\right|^2,\\ &\color{green}{\mathcal{T} = \frac{J_T}{J_I} = |A_3|^2} = \left|\frac{2 e^{-ik_3 L}k_1 \kappa_2}{(k_1+k_3)\kappa_2 \cosh(L \kappa_2) + i(- k_1 k_3+\kappa_2^2)\sinh(L \kappa_2)}\right|^2, \end{align} \]

\[ \begin{equation} \color{green}{\mathcal{T}} = \left|\frac{2 e^{-ik_3 L}k_1 \kappa_2}{(k_1+k_3)\kappa_2 \cosh(L \kappa_2) + i(- k_1 k_3+\kappa_2^2)\sinh(L \kappa_2)}\right|^2, \end{equation} \]

for thick and/or tall barriers (\(L\kappa_2 \gg 1\)) (problem)

\[\begin{equation} \mathcal{T} = |A_3|^2 \approx 4 e^{-2\kappa_2 L}\left[1+\left(\frac{\kappa_2^2-k_1^2}{2k_1\kappa_2}\right)^2\right]^{-1}. \end{equation} \]

for thin barriers and/or energies right below the barrier (\(L\kappa_2 \ll 1\)) (problem) \[ \begin{equation} \mathcal{T} = |A_3|^2 \approx 1- (L\kappa_2)^2\left[1+\left(\frac{\kappa_2^2-k_1^2}{2k_1\kappa_2}\right)^2\right]. \end{equation} \]

Take-home messages

• quantum tunnelling across potential barriers emerges naturally from the Schrodinger equation,
• thin, short obstacles are no match for quantum tunnelling

\[ \begin{equation} \mathcal{T}\approx 1 ~\text{for}~ L \kappa_2 \ll 1, \end{equation} \]

• transmission of the wavefunction drops off exponentially for long, tall barriers \[ \begin{equation} \mathcal{T}\approx \exp(-2L \kappa_2) ~\text{for}~ L \kappa_2 \gg 1. \end{equation} \]

Problems to think over coffee

• what would it take to emulate classical, hard border?
• where does this behaviour come from? it is the quantum feature? a wave phenomenon?
• could one boost the transmission with another obstacle?